Summary: A map \(f\) from the quaternion skew field \(\mathbb{H}\) to itself, can also be thought as a transformation \(f : \mathbb{R}^4 \rightarrow \mathbb{R}^4\). In this manuscript, the Jacobian \(J(f)\) of \(f\) is computed, in the case where \(f\) is a quaternion polynomial. As a consequence, the Cauchy-Riemann equations for \(f\) are derived. It is also shown that the Jacobian determinant of \(f\) is non negative over \(\mathbb{H}\). The above commensurates well with the theory of analytic functions of one complex variable.